Quadrangulating a Mesh using Laplacian Eigenvectors

Resampling raw surface meshes is one of the most fundamental operations used by nearly all digital geometry processing systems. The vast majority of work in the past has focused on triangular remeshing; the equally important problem of resampling surfaces with quadrilaterals has remained largely unaddressed. Despite the relative lack of attention, the need for quality quadrangular resampling methods is of central importance in a number of important areas of graphics. Quadrilaterals are the preferred primitive in many cases, such as Catmull-Clark subdivision surfaces, fluid dynamics, and texture atlasing. We propose a fundamentally new approach to the problem of quadrangulating manifold polygon meshes. By applying a Morsetheoretic analysis to the eigenvectors of the mesh Laplacian, we have developed an algorithm that can correctly quadrangulate any manifold, no matter its genus. Because of the properties of the Laplacian operator, the resulting quadrangular patches are wellshaped and arise directly from intrinsic properties of the surface, rather than from arbitrary heuristics. We demonstrate that this quadrangulation of the surface provides a base complex that is wellsuited to semi-regular remeshing of the initial surface into a fully conforming mesh composed exclusively of quadrilaterals.